[EM] Approval meets IIA ?

[Ken Johnson]

>Message: 1
>Date: Thu, 04 Mar 2004 00:52:52 +1030
>From: Chris Benham <chrisbenham at bigpond.com>
>...
>
>To my mind, Approval does NOT satisfy Independence of Irrelevant Alternatives (IIA), ...
>
>Initial two candidate election (with ratings out of ten).
>01: A(9)>>B(1)
>99: B(8)>>A(7)
>
>B wins 99 to 1. Now we add a third candidate X ... 
>
>Same voters and initial 2 candidates, but with a third candidate added.
>01:A(9)>>X(2)>B(1)
>99:B(8)>A(7)>>X(1)
>
>A wins 100 to 99. So adding a clone of A, which ALL the voters ranked last, changed A from a 
>1/100 loser to the winner.
>

Chris,

This might be a good argument against Approval, but I don't believe it 
demonstrates a conflict with IIA (as defined by Arrow). There appear to 
be two very different interpretations of IIA, as illustrated by the 
following links:

IIA, definition 1:

"independence of irrelevant alternatives 
<http://www.artpolitic.org/infopedia/in/Independence_of_irrelevant_alternatives.html>: 
if we restrict attention to a subset of options, and apply the social 
choice function only to those, then the result should be compatible with 
the outcome for the whole set of options."
http://www.artpolitic.org/infopedia/ar/Arrow's_impossibility_theorem.html

See also http://cowles.econ.yale.edu/P/cd/d11a/d1123-r.pdf :
"The constitution respects independence of irrelevant alternatives if 
the social relative ranking (higher, lower, or indifferent) of two 
alternatives A and B depends only on their relative ranking by every 
individual."
(There is a more formal and explicit definition in Section 2 of the 
paper. This definition is specific to rank methods, but can be 
generalized to cardinal methods by replacing "their relative ranking" 
with "their rating".)

IIA, definition 2:

"According to IIAC, introducing another candidate into an election 
should not change the winner -- unless that candidate actually wins."
http://www.electionmethods.org/Arrow.htm

Definition 1 relates to how the outcome of a particular election  might 
change based on the vote-counting procedure employed, whereas definition 
2 relates to how the result of different hypothetical elections, held 
under different circumstances with different combinations of candidates, 
might differ (taking into account voter strategy). I believe the 
definition used in Arrow's Theorem is the first (although the second 
might actually be more useful and meaningful).

Applying these definitions to the above example, we start out with the 
CR/Approval ratings:

01: A(9) >> X(2) > B(1)
99: B(8) > A(7) >> X(1)

100 approve A, 99 approve B, and A wins. Following IIA definition 1, we 
ignore X and obtain the result,

01: A(9) >> B(1)
99: B(8) > A(7)

Still, 100 approve A while 99 approve B (as well as A), so there is no 
change. But following definition 2, we consider what the result would be 
if the election were held without candidate X. In this case the 99 
voters who approve of both A and B would insincerely rate B 
"unacceptable" because there are no worse candidates on the ballot:

01: A(9) >> B(1)
99: B(8) >> A(7)  (insincere)

So B wins, demonstrating the susceptibility of Approval to insincere 
voting. However, if the vote split were really 99:1 the B supporters 
would probably know in advance that they are a majority, so they would 
also vote insincerely in the three-way contest,

01: A(9) >> X(2) > B(1)
99: B(8) >> A(7) > X(1) (insincere)

B would win in either case, consistent with both majority rule and 
sincere CR. Thus, it is not clear that this example shows an egregious 
susceptibility to insincerity, and in any case it does not appear to 
demonstrate violation of Arrow's (narrowly defined) IIA premise.

>
>One of my fundamental standards is that a method should perform reasonably when all the voters
>vote sincerely (taking no account of how any other voters might vote). 
>A method should be able to cope with insincerity, but to perform reasonably it definitely shouldn't
>DEPEND on insincerity.
>  
>
This certainly seems like a reasonable criterion (though not one of 
criteria of Arrow's theorem). In the above example sincere Approval 
would select A with or without candidate X, whereas insincere Approval 
would likely select B (as would sincere CR and majority rule). But given 
that A and B both have similar, high ratings (7 and 8), all voters would 
be satisfied with either outcome, so Approval appears to perform 
"reasonably" in this circumstance.

Consider also the following alternative scenario:

49: A(10) > B(9) >> C(0)
51: C(10) > B(9) >> A(0)

Advance polling clearly identifies the two factions' preference ratings, 
but does not clearly indicate who has the majority:

49 +/-3: A(10) > B(9) >> C(0)
51 +/-3: C(10) > B(9) >> A(0)

Thus, under majority rule voters would insincerely rank B highest to 
ensure that one of their preferred candidates wins:

49: B > A > C (insincere)
51: B > C > A (insincere)

In this circumstance, Approval, CR, and insincere majority rule select 
the candidate who has the overwhelming approval of all voters (B), 
whereas sincere majority rule results in an outcome that satisfies only 
about half the voters (C).

A fundamental question with all of these types of comparisions is, by 
what standard can you say that one outcome is more "reasonable" than 
another? Relying on preference rankings to define the standard is 
problematic because the statement "A > B" is so ambiguous. It could mean 
that A has a sincere cardinal rating 1 point higher than B, or 10 points 
higher, or anything in between. In my view, "sincere CR" (e.g. on a 
scale of 0 to 1, no quantization of rating levels) provides a meaningful 
standard for DEFINING the objective of elections, even though the 
complication of strategy makes CR an impractical election method itself.

Ken Johnson

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